Polylogarithms and multiple zeta values from free Rota-Baxter algebras

TL;DR

This paper demonstrates that shuffle algebras for polylogarithms and regularized multiple zeta values are free Rota-Baxter algebras, unifying their shuffle relations through a single series and exploring their double shuffle relations via renormalization.

Contribution

It establishes the freeness of these shuffle algebras as Rota-Baxter algebras and connects their relations through a unified framework, advancing the understanding of MZV algebraic structures.

Findings

Shuffle algebras are free Rota-Baxter algebras with one generator.

Shuffle relations of polylogarithms and MZVs derive from a single series.

Extended double shuffle relations are analyzed via renormalization.

Abstract

We show that the shuffle algebras for polylogarithms and regularized MZVs in the sense of Ihara, Kaneko and Zagier are both free commutative nonunitary Rota-Baxter algebras with one generator. We apply these results to show that the full sets of shuffle relations of polylogarithms and regularized MZVs are derived by a single series. We also take this approach to study the extended double shuffle relations of MZVs by comparing these shuffle relations with the quasi-shuffle relations of the regularized MZVs in our previous approach of MZVs by renormalization.